History of Negative Numbers:



Rebekkah Niles

Mat 3010: History of Mathematics

July 29, 2006

Appendix:

Title Page---------------------------------------------------------------------------------------1

Appendix---------------------------------------------------------------------------------------2

History of Negative Numbers---------------------------------------------------------------3

Objectives--------------------------------------------------------------------------------------4

Standards---------------------------------------------------------------------------------------4

Lesson Plan------------------------------------------------------------------------------------4

Chinese Shang Numerals Guide-----------------------------------shangnumerals

Teacher’s Example for Overhead---------------------------------------shannum2

Teacher’s Answer Guide for Overhead-------------------------------shannum2a

Blank Shang Numeral Practice Sheets------------------------------negnumpract

Negative Numbers Practice Sheet----------------------------------negnumpract2

Negative Numbers Practice Sheet Answers--------------------negnumpractans

Bibliography-----------------------------------------------------------------------------------6

History of Negative Numbers:

Negative numbers may seem about as controversial as zucchini, but what if they did not actually exist? For centuries, mathematicians rejected negative numbers as “absurd.” Oddly enough, however, their first recorded appearance was accepted without argument. In third-century China, negative numbers represented debts owed (Negative Numbers 7). Who could protest that? People in India around 500 CE used negative numbers for similar reasons. Brahmagupta gave rules for working with negative numbers (Negative Numbers 11). Two hundred years later in the Arabic Empire, Al-Khwarzimi used the distributive property working with negative numbers, although he refused to allow negative solutions (Negative Numbers 13). About the five hundred years earlier, Diophantus, a Greek mathematician, had used both symbols and negative numbers in his own calculations, yet refused to allow negative solutions (Negative Numbers 8). What was the problem with negative numbers? Well, say a person has two oranges, and eats three. How many remain? Simply put, it is not physically possible to take away more oranges than a person has! The question of negative numbers remained on the table, growing more controversial as people headed into the second millennium.

By the 1100s, mathematicians worked with negative numbers all over the world. Dhaskara of India even found negative solutions to the quadratic equation, although he disliked them (Negative Numbers 12). Al-Samaw’al put together rules for working with negative numbers (Negative Numbers 15). Fibonacci, also known as Leonardo of Pisa, brought the Arabic-numeral system to Europe, replacing Roman numerals and generally making math easier (Negative Numbers 16). During this time, unknowns began making appearances in equations as symbols; the traditional writing out of math problems in words was changing into the writing of problems as strings of symbols (Negative Numbers 15). This process revived and enhanced itself in the fifteenth and sixteenth centuries. Johann Widman and Francois Viete were both known for using symbolic-based equations (Negative Numbers 17), and Robert Recode of England introduced a version of the “=” sign in the mid-1500s (Negative Numbers 22).

The understanding of negative numbers increased as time progressed. Albert Girard understood and accepted negative numbers as solutions, and also stated the first version of the Fundamental Theory of Algebra (Negative Numbers 25). Ten years after his death, Sir Isaac Newton was born. Newton eventually went on to discover calculus. Colin Maclaurin of Scotland finally created algebraic justifications for the rules dealing with positive and negative numbers in the eighteenth century (Negative Numbers 27). Although some protests against negative numbers continued to be made well into the nineteenth century, Karl Weier pretty much ended the debate by creating a logical foundation for the real number system, including both complex numbers and negative numbers (Negative Numbers 33).

All told, from the first recorded use of negative numbers to Weier’s proofs, it took nearly 1500 hundred years for negative numbers to be fully accepted. Now used today in all facets of math, negative numbers are accepted as solutions, as powers, as coefficients, and just about every other place they can be placed. If there can be a positive, there can be a negative, as long as a person includes the entire complex number system. Yet it took a lot of arguments for negative numbers to catch on. Zucchini certainly didn’t take fifteen hundred years to broach the kitchen table!

Chinese Counting Board Activity

Abstract: In China, scholars used counting boards to add numbers. The rightmost column represented the ones place, the second column to the right the tens place, the third column the hundreds place, and so on. Since Chinese numerals are easy to display as sticks, these numbers could be manipulated easily. Red numerals represented positive numbers, and black numerals represented negatives. When a column had both red and black sticks, one stick from each was removed until no more sticks remained. A stick representing a 5 could only be removed with five ones or another five. After all the sticks had been removed that could be removed, the scholar attempted to put all the numbers on the same line. If the resulting numbers were all red, then the result was positive, but if all the sticks were black, a debt was owed.

Objectives: Students will be able to add negative numbers with positive numbers.

Standards:

North Carolina 6th grade standard 1.01: Develop number sense for negative rational numbers.

NCTM 6th grade standard: Develop meaning for integers and represent and compare quantities with them.

Lesson:

Introduction: Hand out worksheet “Chinese Shang Numerals”

Explain: The ancient Chinese used something very similar to tally-marks to keep track of numbers. The symbol for 1 was a single vertical stick; the symbol for two was two vertical sticks, and so on up to five. The symbol for six was a horizontal stick with a vertical stick dropping down. The numerals in front of you are a basic example of the numerals used.

The Chinese also used a base-ten system. This means that, after nine objects were represented, and a tenth added, a single tally would be added in the next digit of the number. The tens unit place used horizontal numbers, with the hundreds switching back to vertical. This pattern of switching continues.

When a zero is a digit in a number, such as 101, the tens space between the vertical ones is left blank.

Body:

The Chinese used negative numbers as debts, and positive numbers as money received. Positive numbers are represented in red, while negatives are in black. To add positives and negatives, a counting board was used. In the top row, the positive number was listed. In the bottom, the negative number was listed.

To add the two together, the Chinese used movable counters. They added in a column by column basis. Starting in the first column, a tally from the positive row would be removed, and a corresponding tally from the bottom row would also be removed.

---hand out blank sheets and red/black manipulatives (algebra tiles or painted matches will do)—

(on overhead, demonstrate with 1,123+-240. Use clear tiles as the red tiles and black as black)

[pic]

When all the numbers are on the same row, the problem is complete.

In groups of two, practice the following problems:

--hand out Negative Numbers Practice sheet and blank practice sheet---

---Allow students time to work on the sheet, drawing out their final answers on the blank sheet. Afterwards, share the answers using the overhead---

Homework: Create and solve 5 problems using Chinese counting boards. Check your answers with traditional addition.

Bibliography:

Historical Summary of Negative Numbers:

Mathematical Association of America. “Negative Numerals.” ed. Victor Katz & Karen D.

Michalowicz. Historical Modules for the Teaching and Learning of Mathematics.

6-33

Standards:

Education World, Inc. (July 29, 2006) “Standards Grades 5-8.” (2006).

North Carolina Department of Education. (July 29, 2006). “Standard Course of Study: Grade 6.” < >

Shang Numerals:

Mathematical Association of America. “Negative Numerals.” ed. Victor Katz & Karen D.

Michalowicz. Historical Modules for the Teaching and Learning of Mathematics. 35

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